The Hirzebruch-Mumford covolume of some hermitian lattices

Abstract

Let $L=diag(1,1,\dots ,1,-1)$ and $M=diag(1,1,\dots ,1,-2)$ be the lattices of signature $(n,1)$. We consider the groups $\Gamma =SU(L,{\mathcal{O}}_K)$ and ${\Gamma }^{\prime }=SU(M,{\mathcal{O}}_K)$ for an imaginary quadratic field $K=\mathbb{Q}(\sqrt{-d})$ of discriminant $D$ and it's ring of integers ${\mathcal{O}}_K$, $d$ odd and square free. We compute the Hirzebruch-Mumford volume of the factor spaces ${\mathbb{B}}^n/\Gamma$ and ${\mathbb{B}}^n/{\Gamma }^{\prime }$. The result for the factor space ${\mathbb{B}}^n/\Gamma$ is due to Zeltinger, but as we're using it to prove the result for ${\mathbb{B}}^n/{\Gamma }^{\prime }$ and it is hard to find his article, we prove the first result here as well.

References in corpus (1)

• Group schemes and local densities of ramified hermitian lattices in residue characteristic 2 Part I